Question

\[\lim_{x \to \infty} \sqrt{x^2 + cx - x}\] 

Answer 1

\[\lim_{x \to \infty} \left( \sqrt{x^2 + cx - x} \right)\]

It is of the form ∞ - ∞.

Rationalising the numerator: 

\[\lim_{x \to \infty} \left( \sqrt{x^2 + cx} - x \right) \frac{\left( \sqrt{x^2 + cx} + x \right)}{\left( \sqrt{x^2 + cx} + x \right)}\]
\[ = \lim_{x \to \infty} \left( \frac{x^2 + cx - x^2}{\sqrt{x^2 + cx} + x} \right)\]
\[\text{ Dividing the numerator and the denomiator by }x: \]
\[ \lim_{x \to \infty} \left[ \frac{\frac{cx}{x}}{\frac{\sqrt{x^2 + cx} + x}{x}} \right]\]
\[ = \lim_{x \to \infty} \left[ \frac{c}{\sqrt{1 + \frac{c}{x}} + 1} \right]\]
\[\text{ When } x \to \infty , \text{ then } \frac{1}{x} \to 0 . \]
\[ \Rightarrow \frac{c}{\sqrt{1} + 1}\]
\[ = \frac{c}{2}\]